1. It would depend on the heater because some draw their intake air from the bottom, and WHEN the heater settled into the foam, and it would, the combustion air would be cut off.
2. ANY heater will compress it because of the weight
3. WHY would you want to anyway? There is little, if any, heat loss downward from the heater.

Does it cause any problems to put a block of insulating foam under a 50-gallon water heater before installing it? If not, what density is required?

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I doubt the answer will differ, but what type of water heater, electric or gas?

For gas, I don't see that it would serve much if any purpose and depending on what is selected there could be major problems. There are two standard designs/manufacturers of traditional atmospheric vent gas water heater. One has its combustion air draw on the bottom. Therefore you can't cover that. You could rest the legs on some sort of insulating isolation pads, but the area under the water heater should not have combustible materials. (There is non-foam insulation surrounding the walls of the combustion chamber of the one I have, but this does not appear to be the case of the base itself.)

Even if you thermally isolate the feet of a gas water heater from contact with the ground, they still will behave as "fins" from a heat transfer perspective so I doubt the overall losses from them would change much unless you actually made an insulated boot for each foot.

For electric I suspect the bottom head of the water heater is already well insulated, I can't think of any reason it shouldn't be, whereas gas is more challenging. Since electric water heaters have thicker insulation on average (R16 seems to be the starting point vs. R8 for the starting point for gas) there is much less potential for savings for adding insulation to them. The thick insulation's impact is reflected in the very high efficiency factors for electric: for example, State's 50 gal electrics in last year's product guide were 0.91 for R16 (2"), 0.93 for R20 (2.5"), 0.95 for R24 (3").

In the Seattle area, we install electric water heaters on foam pads.
We buy them from the supplier that supplies the water heaters.

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Is that a carryover from years past (days of thinner insulation), or is it relatively recent? If the manufacturers don't insulate the bottom head as much as the walls and top, then this could make some difference, particularly since this end is actually contacting a solid surface such as concrete and would have a higher conductivity.

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At least ten years.
It prevents the tank from rusting on the bottom too.

If you have ever slept on snow, and used a closed cell sleeping pad, you would know how it works.

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I actually wondered if it might help prevent rusting of the metal insulation jacket like that. Having that insulating layer next to it should keep the surface much warmer and drier even for today's thicker insulation.

Haven't slept on snow, but use my Therm-a-rest camping a few times a year...not often enough. With kids' sporting events/coaching etc. it seems like we can never get away in prime camping season.

It's all I could do to haul my pack up there.
Our water heating was a small gas stove.
Going to the top was good for me ego, but I prefer back packing where there are fish, or taking lifts and doing some serious downhill skiing.

I was envious of the guys going back down on skis and snoboards.

The funny thing, was all the younger kids that would see us old guys, and they would kill themselves to pass us.
Then while they were panting their lungs out, we would slowly move by them.
Finally, one of the kids asked if he could follow us, and go up at our pace.

And then, when we were near Muir, they made their bust of speed.
Hey, you gotta beat old guys when you can.

Looks like this is an OK idea for electric and maybe sealed combustion gas too. But I'd still like to know what the compressive strength of the foam should be.

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The feet of the tank should be carrying the weight, not the foam, so the compressive strength is irrelevant.

But if you insist, consider that a 50gallon tank has ~420lbs of water in it, plus it's dry weight- call it 600lbs. Most 50 gallon tanks are ~20-24" in diameter. For a 20" diameter version, that's ~2.2 square feet. For the foam to carry the load, it needs to be able to handle ~1.9psi or ~275t or more with MINIMAL deformation. XPS handles 25psi with 10% deformation, EPS handles 10psi/10%. They'll all handle it in a static load condition (on a concrete slab, perhaps), but under the dynamics of a wooden floor, or anything in an earthquake or even a street with truck traffic you shouldn't be counting on the foam as a primary structural element.

The feet of the tank should be carrying the weight, not the foam, so the compressive strength is irrelevant.

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Correct me if I'm wrong, but in order for the feet to carry the weight, that force has to be distributed over the surface of any foam beneath (such as by sitting on a plate or pan.) Otherwise the feet will just cut through the foam. Are the electrics actually on feet keeping them off the ground? (Or perhaps just button sized nubs?) I haven't examined them closely in recent years.

Correct me if I'm wrong, but in order for the feet to carry the weight, that force has to be distributed over the surface of any foam beneath (such as by sitting on a plate or pan.) Otherwise the feet will just cut through the foam. Are the electrics actually on feet keeping them off the ground? (Or perhaps just button sized nubs?) I haven't examined them closely in recent years.

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The feet have to extend to something structural, not resting on foam. If it's a flat-bottomed beast with mere dimpled sheet metal for feet it calls for building some. The last tank heater I owned was ca. 1982, had about 1.5-2" of clearance under it between some fairly substantial welded on feet, but I'm sure they're everything from dead-flat to fairly tall. I've seen a number of them with 2-3" structural rings beneath them for supporting the tank with some clearance,but I have no idea what/if there's a standard (I expect not.) Clearly it's easy to exceed 25psi many times over if it's a 3 metal feet or a stamped angle-iron ring holding up 600lbs. Looks like electric tanks often come with minimal clearance, whereas gas versions build in the necessary clearances for combusion air intake & burner heat issues: